Jacobian

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Its importance lies in the fact that it represents the best linear approximation to a differentiable function near a given point.

The Jacobian matrix is named after the mathematician Carl Gustav Jacobi; the term "Jacobian" is pronounced as "yah-KO-bee-un".

Suppose F : Rⁿ → R^m is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y₁(x₁,...,x_n), ..., y_m(x₁,...,x_n). The partial derivatives of all these functions (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix of F, as follows:

<math>\begin{bmatrix} \partial y_1 / \partial x_1 & \cdots & \partial y_1 / \partial x_n \\ \vdots & \cdots & \vdots \\ \partial y_m / \partial x_1 & \cdots & \partial y_m / \partial x_n \end{bmatrix} </math>

This matrix is denoted by

<math>J_F(x_1,\ldots,x_n) \qquad \mbox{or by}\qquad \frac{\partial(y_1,\ldots,y_m)}{\partial(x_1,\ldots,x_n)}</math>

The i-th row of this matrix is given by the gradient of the function y_i for i=1,...,m.

If p is a point in Rⁿ and F is differentiable at p, then its derivative is given by J_F(p) (and this is the easiest way to compute said derivative). In this case, the linear map described by J_F(p) is the best linear approximation of F near the point p, in the sense that

<math>F(\mathbf{x}) \approx F(\mathbf{p}) + J_F(\mathbf{p})\cdot (\mathbf{x}-\mathbf{p})</math>

for x close to p.

Example The Jacobian matrix of the function F : R³ → R⁴ with components:

y₁ = x₁

y₂ = 5x₃

y₃ = 4(x₂)² - 2x₃

y₄ = x₃sin(x₁)

is:

<math>J_F(x_1,x_2,x_3) =\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8x_2 & -2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end{bmatrix} </math>

Jacobian determinant If m = n, then F is a function from n-space to n-space and the Jacobi matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant.

The Jacobian determinant at a given point gives important information about the behavior of F near that point. For instance, the continuously differentiable function F is invertible[?] near p if and only if the Jacobian determinant at p is non-zero. This is the inverse function theorem[?]. Furthermore, if the Jacobian determinant at p is positive, then F preserves orientation near p; if it is negative, F reverses orientation. The absolute value of the Jacobian determinant at p gives us the factor by which the function F expands or shrinks volumes near p; this is why it occurs in the general substitution rule.

Example The Jacobian determinant of the function F : R³ → R³ with components

y₁ = 5x₂

y₂ = 4(x₁)² - 2sin(x₂x₃)

y₃ = x₂x₃

is:

<math>\begin{vmatrix} 0 & 5 & 0 \\ 8x_1 & -2x_3\cos(x_2 x_3) & -2x_2\cos(x_2 x_3) \\ 0 & x_3 & x_2 \end{vmatrix}=-8x_1\cdot\begin{vmatrix} 5 & 0\\ x_3&x_2\end{vmatrix}=-40x_1 x_2</math>

From this we see that F reverses orientation near those points where x₁ and x₂ have the same sign; the function is locally invertible everywhere except near points where x₁=0 or x₂=0. If you start with a tiny object around the point (1,1,1) and apply F to that object, you will get an object set with about 40 times the volume of the original one.